When Inefficient Methods Have Greater Outcomes
I love watching kindergarteners do math. They have such wonder and enjoyment as they play with objects and numbers. The fact that they are new to the school environment means they have yet to learn and use more traditional ways of ‘doing math’. For example, they may not be using five frames to organize and count objects as they begin to understand numbers, but kindergarteners do have a mathematical way about them. They sort objects, see patterns, and are fantastic at recognizing more! (Just give their sibling one more M&M for snack)
Not long ago, I was invited in to teach Mrs. A’s kindergarteners after I had recently shared a math and literature lesson with a group of teachers. The teacher and I were excited to see how students were thinking about counting and it was the perfect opportunity to let students lead with their own methods of counting. For this lesson, we read Chrysanthemum, written by Kevin Henkes, a fantastic book that many teachers use for its lessons on kindness and honoring differences. Marilyn Burns, math educational guru, has always spoken about the power of using literature in math class. She has a lesson on her website, Letters in Our Names, where students read Chrysanthemum and explore the number of letters in first names. With younger students, I use this page from the book.
Chrysanthemum by Kevin Henkes
Inspired by another lesson from Marilyn Burns, I asked the students to consider the number of pockets that Chrysanthemum wore on her outfit to school that day. Then, I posed another question: How many pockets do you think we all have in the classroom today?
Right away, students started searching their own outfits for pockets. I modeled finding the number of pockets in my own outfit by placing a connecting cube in each pocket, removing the cubes, and connecting them into a stack. I asked students how I could find the total number of my ‘pockets’ (as represented by the cubes). Once we were satisfied with how to find the number of pockets, students were sent off to tables to grab cubes, place them in their own pockets, and remove and stack their cubes. Students stood around the carpet to do some comparisons.
Me: “Turn to a neighbor and compare your stack. What do you notice?”
D: “I have more.”
A: “Mine is the same as hers.”
T: “My stack is long. I have a lot of pockets on me.”
M: “All of us (pointing to herself and two others) have the same size stack.”
I then went back to the original question and asked the students how many pockets they thought would be in our whole classroom. Students were very excited and started throwing out numbers. I started recording student estimates as written numerals. Most students picked numbers from 20-50. A few students picked numbers around 100 and 2 students picked made up numbers that sounded large (eleventy-hundred). It was a great opportunity to assess how students are thinking about larger numbers in their world.
I then asked students to think about how we could count all of the stacks. After independent think time and an opportunity to share with their neighbor, students came up with three different ways.
Count all of them
Put them in groups
Count by tens
Students had been counting by tens as their warm-up routine so it made sense that they offered that suggestion. We decided to count by tens.
Me: “Well I have 5 cubes in my stack. How can I count by tens?”
T: “I have 5 too! I could put my stack on yours.”
O: “Yeah. Then we have a stack of 10.”
Me: “Ok. Let’s stack these together and set them in the middle of the carpet so we can see. Now what?”
S: “Oh! M and I both have 5 and put them together and there’s 10.”
Me: “Go ahead and then place them on the carpet next to mine.”
B: “I have 3 and she has 1 and we can put them together. We only have 4.”
D: “I have 6 and that could go with your 4.”
Me: “Why can we put those three stacks together?”
B: “Well 4 and 6 more make a group of 10. You can see it on our ten frames over there.” (pointing to previous work they had been doing)
The class continued grouping stacks by 10 and when we finished, we counted the groups and got exactly 90. The students were delighted and surprised by the large number.
I was pleased to see that students demonstrated problem solving skills and worked together to find a solution that made sense to them. I also wondered about students who suggested that we count all of the cubes. I wasn’t certain that they trusted counting by tens would give them the same result as counting them all.
Me: “Some of you suggested we count all of the cubes. If we were to line up our stacks into a long row and count them all, would we still get 90?”
After thinking for a moment, the room was divided with most students saying that the number would change and a few students certain that the number would still be 90.
N: “It would be the same.”
Me: “Why do you think so?”
N: “We didn’t add any or take any away. It’s the same number.”
Me: “What do others think about N’s statement?”
A: “I’m not sure. I think it’s more.”
I could see many students with confused looks on their faces. This was a perfect moment of disequilibrium where students are grappling with the difference between counting by ones and tens. On one hand, it’s more efficient to get students to count by tens, but in the end, if they don’t trust that counting by ones and then by tens produces the same result, their understanding of counting and quantity will be shaky.
I then modeled lining up the cubes into one long stack. I verified that we were only using the stacks that we previously counted but that we will now count them all to see if we still get 90. I pointed to each cube as we counted and stopped a couple times to verify the number we were on. When we reached the end and had counted all 90 cubes, it was fascinating to see the amazement on students’ faces who hadn’t been sure. As a class, many students were able to verbalize why counting by tens and then ones gave us the same number.
This experience continues to remind me of the importance of honoring student generated methods and inefficient strategies while students are learning. I recently read a post from Pam Harris’ podcast, author of Math is Figure-Out-Able where she commented, “Algorithms are amazing historical achievements. They’re just not very good teaching tools.” While students in kindergarten are rarely learning algorithms, it’s a reminder that the students themselves need to uncover big mathematical ideas through experience, conversation, and play for a deep understanding. The students have to internalize why something works, and as a teacher, I have to know when I should provide more opportunities and in what ways.
I wonder if students were to sit in the ‘play’ or ‘wonder’ of math longer versus moving quickly into formal methods if it might benefit them in the long run? What are some other examples you can think of?
Take care,
Holly
Burns, Marilyn. MB Math. 2015, https://marilynburnsmath.com/data/chrysanthemum-an-oldie-but-goodie/. Accessed April 2, 2024.
“Look Fors for Students.” Math is Figure-Out-Able with Pam Harris, March 19, 2024, https://podcast.mathisfigureoutable.com/1062400/14582459
Henkes, Kevin. Chrysanthemum. New York, Harper Collins, 1991.
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